module stringlengths 16 90 | startPos dict | endPos dict | goals listlengths 0 96 | ppTac stringlengths 1 14.5k | elaborator stringclasses 366
values | kind stringclasses 370
values |
|---|---|---|---|---|---|---|
Mathlib.Analysis.Asymptotics.ExpGrowth | {
"line": 238,
"column": 2
} | {
"line": 240,
"column": 91
} | [
{
"pp": "u : ℕ → ℝ≥0∞\n⊢ expGrowthInf u⁻¹ = -expGrowthSup u",
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
"EReal.instDivInvMonoid",
"DivInvMonoid.toInv",
"ExpGrowth.expGrowthInf",
"Pi.instNeg",
"instHDiv",
"HMul.hMul",
"Filter.liminf",
... | rw [expGrowthSup, ← liminf_neg]
refine liminf_congr (Eventually.of_forall fun n ↦ ?_)
rw [Pi.neg_apply, Pi.inv_apply, div_eq_mul_inv, div_eq_mul_inv, ← EReal.neg_mul, log_inv] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Asymptotics.ExpGrowth | {
"line": 238,
"column": 2
} | {
"line": 240,
"column": 91
} | [
{
"pp": "u : ℕ → ℝ≥0∞\n⊢ expGrowthInf u⁻¹ = -expGrowthSup u",
"usedConstants": [
"instAddCommMonoidWithOneEReal",
"Eq.mpr",
"EReal.instDivInvMonoid",
"DivInvMonoid.toInv",
"ExpGrowth.expGrowthInf",
"Pi.instNeg",
"instHDiv",
"HMul.hMul",
"Filter.liminf",
... | rw [expGrowthSup, ← liminf_neg]
refine liminf_congr (Eventually.of_forall fun n ↦ ?_)
rw [Pi.neg_apply, Pi.inv_apply, div_eq_mul_inv, div_eq_mul_inv, ← EReal.neg_mul, log_inv] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Asymptotics.SpecificAsymptotics | {
"line": 198,
"column": 4
} | {
"line": 198,
"column": 47
} | [
{
"pp": "case h\nE : Type u_1\ninst✝¹ : NormedAddCommGroup E\ninst✝ : NormedSpace ℝ E\nu : ℕ → E\nl : E\nh : Tendsto u atTop (𝓝 l)\nthis : (fun n ↦ ∑ i ∈ range n, (u i - l)) =o[atTop] fun n ↦ ↑n\nn : ℕ\nnpos : n ∈ Set.Ici 1\nnposℝ : 0 < ↑n\n⊢ (↑n)⁻¹ • ↑n = 1",
"usedConstants": [
"Eq.mpr",
"Grou... | rw [smul_eq_mul, inv_mul_cancel₀ nposℝ.ne'] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Asymptotics.ExpGrowth | {
"line": 310,
"column": 6
} | {
"line": 310,
"column": 18
} | [
{
"pp": "ι : Type u_1\ninst✝ : Finite ι\nu : ι → ℕ → ℝ≥0∞\n⊢ expGrowthInf (⨅ i, u i) = ⨅ i, expGrowthInf (u i)",
"usedConstants": [
"Eq.mpr",
"instInfSetEReal",
"ExpGrowth.expGrowthInf",
"iInf",
"congrArg",
"CompletelyDistribLattice.toCompleteLattice",
"Set.univ",
... | ← iInf_univ, | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.MeasureTheory.Measure.Decomposition.Hahn | {
"line": 117,
"column": 71
} | {
"line": 118,
"column": 77
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), μ s ≠ ∞\nhν : ∀ (s : Set α), ν s ≠ ∞\nto_nnreal_μ : ∀ (... | by
rw [f_succ _ _ hmn, d_split (f m n) (e (n + 1)) (he₁ _), add_assoc] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 223,
"column": 2
} | {
"line": 225,
"column": 67
} | [
{
"pp": "case neg.inr\nα : Type u_1\nι : Type u_2\ninst✝ : PseudoMetricSpace α\nt : Set ι\nx : ι → α\nr : ι → ℝ\nR : ℝ\nhr : ∀ a ∈ t, r a ≤ R\nτ : ℝ\nhτ : 3 < τ\nh✝ : t.Nonempty\nht : ∃ a ∈ t, 0 < r a\nt' : Set ι := {a | a ∈ t ∧ 0 < r a}\nu : Set ι\nut' : u ⊆ t'\nu_disj : u.PairwiseDisjoint fun a ↦ ball (x a) (... | · rcases ht with ⟨b, rb⟩
rcases A b ⟨rb.1, rb.2⟩ with ⟨c, cu, _⟩
exact ⟨c, cu, by simp only [ball_eq_empty.2 h'a, empty_subset]⟩ | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 294,
"column": 2
} | {
"line": 294,
"column": 52
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁴ : PseudoMetricSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : SecondCountableTopology α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set α\nt : Set ι\nC : ℝ≥0\nr : ι → ℝ\nc : ι → α\nB : ι → Set α\nhB : ∀ a ∈ t, B a ⊆ closedBall ... | let v := { a ∈ u | (B a ∩ ball x (R x)).Nonempty } | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticLet___1 | Lean.Parser.Tactic.tacticLet__ |
Mathlib.MeasureTheory.Measure.Decomposition.Hahn | {
"line": 127,
"column": 80
} | {
"line": 128,
"column": 40
} | [
{
"pp": "α : Type u_1\nmα : MeasurableSpace α\nμ ν : Measure α\ninst✝¹ : IsFiniteMeasure μ\ninst✝ : IsFiniteMeasure ν\nd : Set α → ℝ := fun s ↦ ↑(μ s).toNNReal - ↑(ν s).toNNReal\nc : Set ℝ := d '' {s | MeasurableSet s}\nγ : ℝ := sSup c\nhμ : ∀ (s : Set α), μ s ≠ ∞\nhν : ∀ (s : Set α), ν s ≠ ∞\nto_nnreal_μ : ∀ (... | by
simpa only [mul_zero, tsub_zero] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 374,
"column": 6
} | {
"line": 374,
"column": 66
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁴ : PseudoMetricSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : SecondCountableTopology α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set α\nt : Set ι\nC : ℝ≥0\nr : ι → ℝ\nc : ι → α\nB : ι → Set α\nhB : ∀ a ∈ t, B a ⊆ closedBall ... | exact (mem_diff _).2 ⟨mem_of_mem_inter_right hz, z_notmem_k⟩ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Covering.Vitali | {
"line": 408,
"column": 6
} | {
"line": 408,
"column": 34
} | [
{
"pp": "α : Type u_1\nι : Type u_2\ninst✝⁴ : PseudoMetricSpace α\ninst✝³ : MeasurableSpace α\ninst✝² : OpensMeasurableSpace α\ninst✝¹ : SecondCountableTopology α\nμ : Measure α\ninst✝ : IsLocallyFiniteMeasure μ\ns : Set α\nt : Set ι\nC : ℝ≥0\nr : ι → ℝ\nc : ι → α\nB : ι → Set α\nhB : ∀ a ∈ t, B a ⊆ closedBall ... | rcases ab with ⟨e, ⟨ea, eb⟩⟩ | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalRCases | Lean.Parser.Tactic.rcases |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 633,
"column": 2
} | {
"line": 633,
"column": 59
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nν μ : Measure α\ninst✝¹ : IsFiniteMeasure ν\ninst✝ : ν.HaveLebesgueDecomposition μ\nr : ℝ≥0∞\nhr : r ≠ ∞\nh : (r.toNNReal • ν).rnDeriv μ =ᶠ[ae μ] r.toNNReal • ν.rnDeriv μ\n⊢ (r • ν).rnDeriv μ =ᶠ[ae μ] r • ν.rnDeriv μ",
"usedConstants": [
"ENNReal.coe_toNNR... | simpa [ENNReal.smul_def, ENNReal.coe_toNNReal hr] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Analysis.Calculus.Monotone | {
"line": 61,
"column": 2
} | {
"line": 61,
"column": 19
} | [
{
"pp": "f : ℝ → ℝ\nx a c d : ℝ\nl : Filter ℝ\nhl : l ≤ 𝓝[≠] x\nhf : Tendsto (fun y ↦ (f y - d) / (y - x)) l (𝓝 a)\nh' : Tendsto (fun y ↦ y + c * (y - x) ^ 2) l l\nL : Tendsto (fun y ↦ (y + c * (y - x) ^ 2 - x) / (y - x)) l (𝓝 1)\nZ :\n Tendsto\n (fun x_1 ↦\n ((fun y ↦ (f y - d) / (y - x)) ∘ fun y ↦... | rw [mul_one] at Z | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.MeasureTheory.Measure.Decomposition.Lebesgue | {
"line": 1006,
"column": 2
} | {
"line": 1006,
"column": 59
} | [
{
"pp": "α : Type u_1\nm : MeasurableSpace α\nν μ : Measure α\ninst✝¹ : SigmaFinite ν\ninst✝ : SigmaFinite μ\nr : ℝ≥0∞\nhr : r ≠ ∞\nh : (r.toNNReal • ν).rnDeriv μ =ᶠ[ae μ] r.toNNReal • ν.rnDeriv μ\n⊢ (r • ν).rnDeriv μ =ᶠ[ae μ] r • ν.rnDeriv μ",
"usedConstants": [
"ENNReal.coe_toNNReal",
"Measure... | simpa [ENNReal.smul_def, ENNReal.coe_toNNReal hr] using h | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 862,
"column": 2
} | {
"line": 863,
"column": 56
} | [
{
"pp": "α : Type u_1\ninst✝⁴ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝³ : NormedAddCommGroup E\ninst✝² : SecondCountableTopology α\ninst✝¹ : BorelSpace α\ninst✝ : IsLocallyFiniteMeasure μ\nf : α → E\nhf : LocallyIntegrable f μ\nu : ℕ → Set α\nu_open :... | filter_upwards [v.eventually_filterAt_subset_of_nhds ((u_open n).mem_nhds hn),
v.eventually_filterAt_measurableSet x] with a ha h'a | Mathlib.Tactic._aux_Mathlib_Order_Filter_Defs___elabRules_Mathlib_Tactic_filterUpwards_1 | Mathlib.Tactic.filterUpwards |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 892,
"column": 2
} | {
"line": 892,
"column": 40
} | [
{
"pp": "case h\nα : Type u_1\ninst✝⁶ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : BorelSpace α\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf ... | rw [tendsto_iff_norm_sub_tendsto_zero] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Topology.EMetricSpace.BoundedVariation | {
"line": 118,
"column": 82
} | {
"line": 119,
"column": 98
} | [
{
"pp": "α : Type u_1\ninst✝¹ : LinearOrder α\nE : Type u_2\ninst✝ : PseudoEMetricSpace E\nf : α → E\ns : Set α\nn : ℕ\nu : ℕ → α\nhu : MonotoneOn u (Iic n)\nus : ∀ i ≤ n, u i ∈ s\n⊢ ∑ i ∈ Finset.range n, edist (f (u (i + 1))) (f (u i)) ≤ eVariationOn f s",
"usedConstants": [
"eVariationOn.sum_le_of_m... | by
simpa using sum_le_of_monotoneOn_Icc (m := 0) (hu.mono Icc_subset_Iic_self) fun i hi ↦ us i hi.2 | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 899,
"column": 4
} | {
"line": 899,
"column": 42
} | [
{
"pp": "case h\nα : Type u_1\ninst✝⁶ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : BorelSpace α\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf ... | exact norm_integral_le_integral_norm _ | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 899,
"column": 4
} | {
"line": 899,
"column": 42
} | [
{
"pp": "case h\nα : Type u_1\ninst✝⁶ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : BorelSpace α\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf ... | exact norm_integral_le_integral_norm _ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.MeasureTheory.Covering.Differentiation | {
"line": 899,
"column": 4
} | {
"line": 899,
"column": 42
} | [
{
"pp": "case h\nα : Type u_1\ninst✝⁶ : PseudoMetricSpace α\nm0 : MeasurableSpace α\nμ : Measure α\nv : VitaliFamily μ\nE : Type u_2\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : SecondCountableTopology α\ninst✝³ : BorelSpace α\ninst✝² : IsLocallyFiniteMeasure μ\ninst✝¹ : NormedSpace ℝ E\ninst✝ : CompleteSpace E\nf ... | exact norm_integral_le_integral_norm _ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | {
"line": 146,
"column": 6
} | {
"line": 146,
"column": 39
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝¹⁹ : Semifield R\ninst✝¹⁸ : StarRing R\ninst✝¹⁷ : MetricSpace R\ninst✝¹⁶ : IsTopologicalSemiring R\ninst✝¹⁵ : ContinuousStar R\ninst✝¹⁴ : Semifield S\ninst✝¹³ : StarRing S\ninst✝¹² : MetricSpace S\ninst✝¹¹ : IsTopologicalSemiring S\ninst✝¹⁰... | simp [((h a).mp ha).2.left_inv _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Restrict | {
"line": 311,
"column": 6
} | {
"line": 311,
"column": 39
} | [
{
"pp": "R : Type u_1\nS : Type u_2\nA : Type u_3\np q : A → Prop\ninst✝²³ : Semifield R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : Field S\ninst✝¹⁷ : StarRing S\ninst✝¹⁶ : MetricSpace S\ninst✝¹⁵ : IsTopologicalRing S\ninst✝¹⁴ : Conti... | simp [((h a).mp ha).2.left_inv _] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Unital | {
"line": 627,
"column": 20
} | {
"line": 627,
"column": 33
} | [
{
"pp": "R : Type u_1\nA : Type u_2\np : A → Prop\ninst✝⁹ : CommSemiring R\ninst✝⁸ : StarRing R\ninst✝⁷ : MetricSpace R\ninst✝⁶ : IsTopologicalSemiring R\ninst✝⁵ : ContinuousStar R\ninst✝⁴ : TopologicalSpace A\ninst✝³ : Ring A\ninst✝² : StarRing A\ninst✝¹ : Algebra R A\ninstCFC : ContinuousFunctionalCalculus R ... | cfc_pow_id .. | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 233,
"column": 2
} | {
"line": 233,
"column": 47
} | [
{
"pp": "case h\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A\n... | simp only [cfcₙ_apply (f i) a (hf i) (hf0 i)] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.NonUnital | {
"line": 293,
"column": 4
} | {
"line": 293,
"column": 27
} | [
{
"pp": "case neg\nR : Type u_1\nA : Type u_2\np : A → Prop\ninst✝¹¹ : CommSemiring R\ninst✝¹⁰ : Nontrivial R\ninst✝⁹ : StarRing R\ninst✝⁸ : MetricSpace R\ninst✝⁷ : IsTopologicalSemiring R\ninst✝⁶ : ContinuousStar R\ninst✝⁵ : NonUnitalRing A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : Module R A... | obtain (h | h | h) := h | _private.Lean.Elab.Tactic.RCases.0.Lean.Elab.Tactic.RCases.evalObtain | Lean.Parser.Tactic.obtain |
Mathlib.Analysis.Real.Spectrum | {
"line": 38,
"column": 65
} | {
"line": 39,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\na : A\nha : SpectrumRestricts a ⇑ContinuousMap.realToNNReal\nr : ℝ≥0\n⊢ (∀ x ∈ spectrum ℝ≥0 a, r ≤ x) ↔ ∀ x ∈ spectrum ℝ a, ↑r ≤ x",
"usedConstants": [
"NNReal.instTopologicalSpace",
"NNReal.instCommSemiring",
"Real.instLE",
... | by
simp [← ha.algebraMap_image] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Spectrum | {
"line": 43,
"column": 65
} | {
"line": 44,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\na : A\nha : SpectrumRestricts a ⇑ContinuousMap.realToNNReal\nr : ℝ≥0\n⊢ (∀ x ∈ spectrum ℝ≥0 a, r < x) ↔ ∀ x ∈ spectrum ℝ a, ↑r < x",
"usedConstants": [
"NNReal.instTopologicalSpace",
"NNReal.instCommSemiring",
"Real",
"Preo... | by
simp [← ha.algebraMap_image] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Spectrum | {
"line": 48,
"column": 65
} | {
"line": 49,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\na : A\nha : SpectrumRestricts a ⇑ContinuousMap.realToNNReal\nr : ℝ≥0\n⊢ (∀ x ∈ spectrum ℝ≥0 a, x ≤ r) ↔ ∀ x ∈ spectrum ℝ a, x ≤ ↑r",
"usedConstants": [
"NNReal.instTopologicalSpace",
"NNReal.instCommSemiring",
"Real.instLE",
... | by
simp [← ha.algebraMap_image] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Spectrum | {
"line": 53,
"column": 65
} | {
"line": 54,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝¹ : Ring A\ninst✝ : Algebra ℝ A\na : A\nha : SpectrumRestricts a ⇑ContinuousMap.realToNNReal\nr : ℝ≥0\n⊢ (∀ x ∈ spectrum ℝ≥0 a, x < r) ↔ ∀ x ∈ spectrum ℝ a, x < ↑r",
"usedConstants": [
"NNReal.instTopologicalSpace",
"NNReal.instCommSemiring",
"Real",
"Preo... | by
simp [← ha.algebraMap_image] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Spectrum | {
"line": 77,
"column": 75
} | {
"line": 78,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝³ : NonUnitalRing A\ninst✝² : Module ℝ A\ninst✝¹ : IsScalarTower ℝ A A\ninst✝ : SMulCommClass ℝ A A\na : A\nha : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal\nr : ℝ≥0\n⊢ (∀ x ∈ σₙ ℝ≥0 a, x ≤ r) ↔ ∀ x ∈ σₙ ℝ a, x ≤ ↑r",
"usedConstants": [
"NNReal.instTopologicalSpace... | by
simp [← ha.algebraMap_image] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Real.Spectrum | {
"line": 82,
"column": 75
} | {
"line": 83,
"column": 30
} | [
{
"pp": "A : Type u_1\ninst✝³ : NonUnitalRing A\ninst✝² : Module ℝ A\ninst✝¹ : IsScalarTower ℝ A A\ninst✝ : SMulCommClass ℝ A A\na : A\nha : QuasispectrumRestricts a ⇑ContinuousMap.realToNNReal\nr : ℝ≥0\n⊢ (∀ x ∈ σₙ ℝ≥0 a, x < r) ↔ ∀ x ∈ σₙ ℝ a, x < ↑r",
"usedConstants": [
"NNReal.instTopologicalSpace... | by
simp [← ha.algebraMap_image] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Algebra.Module.Spaces.CharacterSpace | {
"line": 123,
"column": 43
} | {
"line": 128,
"column": 86
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : CommSemiring 𝕜\ninst✝⁷ : TopologicalSpace 𝕜\ninst✝⁶ : ContinuousAdd 𝕜\ninst✝⁵ : ContinuousConstSMul 𝕜 𝕜\ninst✝⁴ : NonUnitalNonAssocSemiring A\ninst✝³ : TopologicalSpace A\ninst✝² : Module 𝕜 A\ninst✝¹ : T2Space 𝕜\ninst✝ : ContinuousMul 𝕜\n⊢ IsClosed (charact... | by
simp only [union_zero, Set.setOf_forall]
exact
isClosed_iInter fun x =>
isClosed_iInter fun y =>
isClosed_eq (eval_continuous _) <| (eval_continuous _).mul (eval_continuous _) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 110,
"column": 8
} | {
"line": 110,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n| ∀ (u : Set β), IsClosed[inst✝] u → IsClosed[inst✝¹] (f ⁻¹' Iic uᶜ)ᶜ",
"usedConstants": [
"Function.Surjective.forall",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
... | compl_surjective.forall | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 127,
"column": 4
} | {
"line": 127,
"column": 25
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\nthis : ∀ (u : Set β), (f ⁻¹' Iic uᶜ)ᶜ = {x | (f x ∩ u).Nonempty}\n⊢ (∀ (x : α) (u : Set β), (∀ x ∈ u, u ∈ 𝓝 x) → (f x ∩ u).Nonempty → ∀ᶠ (x' : α) in 𝓝 x, (f x' ∩ u).Nonempty) ↔\n ∀ (u : Set β), (∀ x... | forall_comm (α := α), | Mathlib.Tactic._aux_Mathlib_Tactic_SimpRw___elabRules_Mathlib_Tactic_tacticSimp_rw____1 | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 135,
"column": 8
} | {
"line": 135,
"column": 31
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n| ∀ (u : Set β), IsClosed[inst✝] u → IsClosed[inst✝¹] (f ⁻¹' Iic u)",
"usedConstants": [
"Function.Surjective.forall",
"congrArg",
"Compl.compl",
"PartialOrder.toPreorder",
... | compl_surjective.forall | Lean.Elab.Tactic.Conv.evalRewrite | null |
Mathlib.Topology.Semicontinuity.Hemicontinuity | {
"line": 137,
"column": 2
} | {
"line": 137,
"column": 63
} | [
{
"pp": "α : Type u_1\nβ : Type u_2\ninst✝¹ : TopologicalSpace α\ninst✝ : TopologicalSpace β\nf : α → Set β\n⊢ LowerHemicontinuous f ↔ ∀ (x : Set β), IsOpen[inst✝] x → IsOpen[inst✝¹] (f ⁻¹' Iic xᶜ)ᶜ",
"usedConstants": [
"lowerHemicontinuous_iff_isOpen_compl_preimage_Iic_compl"
]
}
] | exact lowerHemicontinuous_iff_isOpen_compl_preimage_Iic_compl | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.Normed.Algebra.Spectrum | {
"line": 572,
"column": 2
} | {
"line": 572,
"column": 21
} | [
{
"pp": "𝕜 : Type u_3\nA : Type u_4\nSA : Type u_5\ninst✝⁵ : NormedRing A\ninst✝⁴ : CompleteSpace A\ninst✝³ : SetLike SA A\ninst✝² : SubringClass SA A\ninst✝¹ : NormedField 𝕜\ninst✝ : NormedAlgebra 𝕜 A\ninstSMulMem : SMulMemClass SA 𝕜 A\nS : SA\nhS : IsClosed[PseudoMetricSpace.toUniformSpace.toTopologicalSp... | rw [frontier_compl] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.Normed.Operator.Completeness | {
"line": 83,
"column": 2
} | {
"line": 84,
"column": 59
} | [
{
"pp": "𝕜 : Type u_1\n𝕜₂ : Type u_2\nF : Type u_4\ninst✝⁶ : NormedAddCommGroup F\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\nE' : Type u_6\ninst✝² : SeminormedAddCommGroup E'\ninst✝¹ : NormedSpace 𝕜 E'\ninst✝ : RingHomIsometric σ₁... | have : Tendsto (fun m => ‖f n x - f m x‖) atTop (𝓝 ‖f n x - g x‖) :=
(tendsto_const_nhds.sub <| tendsto_pi_nhds.1 hg _).norm | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticHave___1 | Lean.Parser.Tactic.tacticHave__ |
Mathlib.Analysis.Normed.Operator.BanachSteinhaus | {
"line": 39,
"column": 2
} | {
"line": 39,
"column": 66
} | [
{
"pp": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric ... | refine (norm_withSeminorms 𝕜₂ F).banach_steinhaus (fun _ x ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Analysis.Normed.Operator.BanachSteinhaus | {
"line": 49,
"column": 2
} | {
"line": 49,
"column": 66
} | [
{
"pp": "E : Type u_1\nF : Type u_2\n𝕜 : Type u_3\n𝕜₂ : Type u_4\ninst✝⁷ : SeminormedAddCommGroup E\ninst✝⁶ : SeminormedAddCommGroup F\ninst✝⁵ : NontriviallyNormedField 𝕜\ninst✝⁴ : NontriviallyNormedField 𝕜₂\ninst✝³ : NormedSpace 𝕜 E\ninst✝² : NormedSpace 𝕜₂ F\nσ₁₂ : 𝕜 →+* 𝕜₂\ninst✝¹ : RingHomIsometric ... | refine (norm_withSeminorms 𝕜₂ F).banach_steinhaus (fun _ x ↦ ?_) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.UrysohnsLemma | {
"line": 189,
"column": 12
} | {
"line": 189,
"column": 76
} | [
{
"pp": "case zero\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nx : X\nc : CU P\n⊢ approx 0 c x ≤ 1",
"usedConstants": [
"Real.instLE",
"Real",
"le_rfl",
"Real.instZero",
"Real.instZeroLEOneClass",
"Compl.compl",
"Membership.mem",
"Set.... | exact indicator_apply_le' (fun _ => le_rfl) fun _ => zero_le_one | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Topology.UrysohnsLemma | {
"line": 189,
"column": 12
} | {
"line": 189,
"column": 76
} | [
{
"pp": "case zero\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nx : X\nc : CU P\n⊢ approx 0 c x ≤ 1",
"usedConstants": [
"Real.instLE",
"Real",
"le_rfl",
"Real.instZero",
"Real.instZeroLEOneClass",
"Compl.compl",
"Membership.mem",
"Set.... | exact indicator_apply_le' (fun _ => le_rfl) fun _ => zero_le_one | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.UrysohnsLemma | {
"line": 189,
"column": 12
} | {
"line": 189,
"column": 76
} | [
{
"pp": "case zero\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nx : X\nc : CU P\n⊢ approx 0 c x ≤ 1",
"usedConstants": [
"Real.instLE",
"Real",
"le_rfl",
"Real.instZero",
"Real.instZeroLEOneClass",
"Compl.compl",
"Membership.mem",
"Set.... | exact indicator_apply_le' (fun _ => le_rfl) fun _ => zero_le_one | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.UrysohnsLemma | {
"line": 303,
"column": 6
} | {
"line": 304,
"column": 77
} | [
{
"pp": "case h\nX : Type u_1\ninst✝ : TopologicalSpace X\nP : Set X → Set X → Prop\nh0 : 0 < 2⁻¹\nh1234 : 2⁻¹ < 3 / 4\nh1 : 3 / 4 < 1\nx : X\nx✝ : True\nn : ℕ\nihn : ∀ (c : CU P), ∀ᶠ (x_1 : X) in 𝓝 x, dist (c.lim x_1) (c.lim x) ≤ (3 / 4) ^ n\nc : CU P\ny : X\nhydl : dist (c.left.right.lim y) (c.left.right.lim... | simp only [pow_succ, c.lim_eq_midpoint, c.left.lim_eq_midpoint,
c.left.left.lim_of_notMem_U _ hxl, c.left.left.lim_of_notMem_U _ hyl] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.LinearAlgebra.AffineSpace.Ordered | {
"line": 158,
"column": 29
} | {
"line": 158,
"column": 71
} | [
{
"pp": "k : Type u_1\nE : Type u_2\ninst✝⁸ : Field k\ninst✝⁷ : LinearOrder k\ninst✝⁶ : IsStrictOrderedRing k\ninst✝⁵ : AddCommGroup E\ninst✝⁴ : PartialOrder E\ninst✝³ : IsOrderedAddMonoid E\ninst✝² : Module k E\ninst✝¹ : IsStrictOrderedModule k E\ninst✝ : PosSMulReflectLE k E\na b : E\nr r' : k\nh : r < r'\n⊢ ... | smul_le_smul_iff_of_pos_left (sub_pos.2 h) | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.UrysohnsLemma | {
"line": 461,
"column": 4
} | {
"line": 461,
"column": 68
} | [
{
"pp": "X : Type u_1\ninst✝² : TopologicalSpace X\ninst✝¹ : RegularSpace X\ninst✝ : LocallyCompactSpace X\ns t : Set X\nhs : IsCompact s\nh's : IsGδ s\nht : IsClosed[inst✝²] t\nhd : Disjoint s t\nU : ℕ → Set X\nU_open : ∀ (n : ℕ), IsOpen[inst✝²] (U n)\nhU : s = ⋂ n, U n\nm : Set X\nm_comp : IsCompact m\nsm : s... | simpa [abs_of_nonneg, (u_pos n).le, (f_range n x).1] using I n x | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 141,
"column": 4
} | {
"line": 142,
"column": 35
} | [
{
"pp": "X : Type u_1\nR : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : Semiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalSemiring R\nf : C(X, R)\n⊢ f ∈ idealOfSet R ∅ ↔ f ∈ ⊥",
"usedConstants": [
"Semiring.toModule",
"congrArg",
"_private.Mathlib.Topology.ContinuousMap.Idea... | simp only [mem_idealOfSet, Set.compl_empty, Set.mem_univ, forall_true_left, Ideal.mem_bot,
DFunLike.ext_iff, zero_apply] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 141,
"column": 4
} | {
"line": 142,
"column": 35
} | [
{
"pp": "X : Type u_1\nR : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : Semiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalSemiring R\nf : C(X, R)\n⊢ f ∈ idealOfSet R ∅ ↔ f ∈ ⊥",
"usedConstants": [
"Semiring.toModule",
"congrArg",
"_private.Mathlib.Topology.ContinuousMap.Idea... | simp only [mem_idealOfSet, Set.compl_empty, Set.mem_univ, forall_true_left, Ideal.mem_bot,
DFunLike.ext_iff, zero_apply] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 141,
"column": 4
} | {
"line": 142,
"column": 35
} | [
{
"pp": "X : Type u_1\nR : Type u_2\ninst✝³ : TopologicalSpace X\ninst✝² : Semiring R\ninst✝¹ : TopologicalSpace R\ninst✝ : IsTopologicalSemiring R\nf : C(X, R)\n⊢ f ∈ idealOfSet R ∅ ↔ f ∈ ⊥",
"usedConstants": [
"Semiring.toModule",
"congrArg",
"_private.Mathlib.Topology.ContinuousMap.Idea... | simp only [mem_idealOfSet, Set.compl_empty, Set.mem_univ, forall_true_left, Ideal.mem_bot,
DFunLike.ext_iff, zero_apply] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 219,
"column": 6
} | {
"line": 219,
"column": 48
} | [
{
"pp": "case neg\nX : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nI : Ideal C(X, 𝕜)\nf : C(X, 𝕜)\nhf : f ∈ idealOfSet 𝕜 (setOfIdeal I)\nε : ℝ≥0\nhε : 0 < ε\nt : Set X := {x | ε / 2 ≤ ‖f x‖₊}\nht : IsClosed t\nhtI : Disjoint t (setOfId... | refine lt_of_le_of_lt ?_ (half_lt_self hε) | Lean.Elab.Tactic.evalRefine | Lean.Parser.Tactic.refine |
Mathlib.Topology.ContinuousMap.Ideals | {
"line": 256,
"column": 6
} | {
"line": 258,
"column": 81
} | [
{
"pp": "case refine_3.refine_2\nX : Type u_1\n𝕜 : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : TopologicalSpace X\ninst✝¹ : CompactSpace X\ninst✝ : T2Space X\nI : Ideal C(X, 𝕜)\nf : C(X, 𝕜)\nhf : f ∈ idealOfSet 𝕜 (setOfIdeal I)\nε : ℝ≥0\nhε : 0 < ε\nt : Set X := {x | ε / 2 ≤ ‖f x‖₊}\nht : IsClosed t\nhtI : Disjo... | · rcases hx with (hx | hx)
· simpa only [zero_add] using add_lt_add_of_lt_of_le (hgt₁ x hx) zero_le'
· simpa only [zero_add] using add_lt_add_of_le_of_lt zero_le' (hgt₂ x hx) | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi | {
"line": 130,
"column": 33
} | {
"line": 130,
"column": 55
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝²³ : CommSemiring R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\ninst✝¹⁶ : Ring A\ninst✝¹⁵ : Ring B\ninst✝¹⁴ : Algebra S A\ninst✝¹³... | rwa [Prod.spectrum_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi | {
"line": 130,
"column": 33
} | {
"line": 130,
"column": 55
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝²³ : CommSemiring R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\ninst✝¹⁶ : Ring A\ninst✝¹⁵ : Ring B\ninst✝¹⁴ : Algebra S A\ninst✝¹³... | rwa [Prod.spectrum_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi | {
"line": 130,
"column": 33
} | {
"line": 130,
"column": 55
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝²³ : CommSemiring R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\ninst✝¹⁶ : Ring A\ninst✝¹⁵ : Ring B\ninst✝¹⁴ : Algebra S A\ninst✝¹³... | rwa [Prod.spectrum_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi | {
"line": 133,
"column": 33
} | {
"line": 133,
"column": 55
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝²³ : CommSemiring R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\ninst✝¹⁶ : Ring A\ninst✝¹⁵ : Ring B\ninst✝¹⁴ : Algebra S A\ninst✝¹³... | rwa [Prod.spectrum_eq] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_tacticRwa___1 | Lean.Parser.Tactic.tacticRwa__ |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi | {
"line": 133,
"column": 33
} | {
"line": 133,
"column": 55
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝²³ : CommSemiring R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\ninst✝¹⁶ : Ring A\ninst✝¹⁵ : Ring B\ninst✝¹⁴ : Algebra S A\ninst✝¹³... | rwa [Prod.spectrum_eq] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Pi | {
"line": 133,
"column": 33
} | {
"line": 133,
"column": 55
} | [
{
"pp": "A : Type u_1\nB : Type u_2\nR : Type u_3\nS : Type u_4\ninst✝²³ : CommSemiring R\ninst✝²² : StarRing R\ninst✝²¹ : MetricSpace R\ninst✝²⁰ : IsTopologicalSemiring R\ninst✝¹⁹ : ContinuousStar R\ninst✝¹⁸ : CommRing S\ninst✝¹⁷ : Algebra R S\ninst✝¹⁶ : Ring A\ninst✝¹⁵ : Ring B\ninst✝¹⁴ : Algebra S A\ninst✝¹³... | rwa [Prod.spectrum_eq] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 98,
"column": 42
} | {
"line": 99,
"column": 65
} | [
{
"pp": "A : Type u_1\ninst✝⁷ : NonUnitalRing A\ninst✝⁶ : Module ℝ A\ninst✝⁵ : SMulCommClass ℝ A A\ninst✝⁴ : IsScalarTower ℝ A A\ninst✝³ : StarRing A\ninst✝² : TopologicalSpace A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : T2Space A\na : A\n⊢ (-a)⁻ = a⁺",
"usedConstants": [
... | by
rw [← eq_comm, ← sub_eq_zero, ← posPart_neg, neg_neg, sub_self] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 97
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : T2Space A\ninst✝ : StarModule ℝ A\nr : ℝ\nhr : 0 ≤ r\na :... | conv_lhs => rw [← neg_neg r, neg_smul, negPart_neg, posPart_smul_of_nonpos (by simpa), neg_neg] | Mathlib.Tactic.Conv._aux_Mathlib_Tactic_Conv___macroRules_Mathlib_Tactic_Conv_convLHS_1 | Mathlib.Tactic.Conv.convLHS |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 97
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : T2Space A\ninst✝ : StarModule ℝ A\nr : ℝ\nhr : 0 ≤ r\na :... | conv_lhs => rw [← neg_neg r, neg_smul, negPart_neg, posPart_smul_of_nonpos (by simpa), neg_neg] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Basic | {
"line": 130,
"column": 2
} | {
"line": 130,
"column": 97
} | [
{
"pp": "A : Type u_1\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : Module ℝ A\ninst✝⁶ : SMulCommClass ℝ A A\ninst✝⁵ : IsScalarTower ℝ A A\ninst✝⁴ : StarRing A\ninst✝³ : TopologicalSpace A\ninst✝² : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝¹ : T2Space A\ninst✝ : StarModule ℝ A\nr : ℝ\nhr : 0 ≤ r\na :... | conv_lhs => rw [← neg_neg r, neg_smul, negPart_neg, posPart_smul_of_nonpos (by simpa), neg_neg] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.PosPart.Isometric | {
"line": 56,
"column": 4
} | {
"line": 56,
"column": 41
} | [
{
"pp": "case inr.a\nA : Type u_1\ninst✝⁵ : NonUnitalNormedRing A\ninst✝⁴ : NormedSpace ℝ A\ninst✝³ : SMulCommClass ℝ A A\ninst✝² : IsScalarTower ℝ A A\ninst✝¹ : StarRing A\ninst✝ : NonUnitalIsometricContinuousFunctionalCalculus ℝ A IsSelfAdjoint\na : A\nha : IsSelfAdjoint a\nx : ℝ\nhx : x ∈ quasispectrum ℝ a\n... | rw [negPart_eq_neg.mpr hx', norm_neg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 129,
"column": 6
} | {
"line": 129,
"column": 19
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : StarRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : Nonne... | cfcₙ_id ℝ≥0 a | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 137,
"column": 24
} | {
"line": 137,
"column": 37
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : StarRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : Nonne... | cfcₙ_id ℝ≥0 a | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.SpecialFunctions.ContinuousFunctionalCalculus.Rpow.Basic | {
"line": 142,
"column": 53
} | {
"line": 142,
"column": 66
} | [
{
"pp": "A : Type u_1\ninst✝⁹ : PartialOrder A\ninst✝⁸ : NonUnitalRing A\ninst✝⁷ : TopologicalSpace A\ninst✝⁶ : StarRing A\ninst✝⁵ : Module ℝ A\ninst✝⁴ : SMulCommClass ℝ A A\ninst✝³ : IsScalarTower ℝ A A\ninst✝² : StarOrderedRing A\ninst✝¹ : NonUnitalContinuousFunctionalCalculus ℝ A IsSelfAdjoint\ninst✝ : Nonne... | cfcₙ_id ℝ≥0 a | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Continuity | {
"line": 671,
"column": 2
} | {
"line": 676,
"column": 50
} | [
{
"pp": "case pos\nR : Type u_2\nA : Type u_3\np : A → Prop\ninst✝¹² : CommSemiring R\ninst✝¹¹ : StarRing R\ninst✝¹⁰ : MetricSpace R\ninst✝⁹ : Nontrivial R\ninst✝⁸ : IsTopologicalSemiring R\ninst✝⁷ : ContinuousStar R\ninst✝⁶ : NonUnitalRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : MetricSpace A\ninst✝³ : Module R A\nin... | · rintro f ⟨hf, hf0⟩ g ⟨hg, hg0⟩
simp only
rw [cfcₙ_apply .., cfcₙ_apply .., isometry_cfcₙHom (R := R) a ha |>.edist_eq]
simp only [ENNReal.coe_one, one_mul]
rw [← ContinuousMapZero.isometry_toContinuousMap.edist_eq,
edist_continuousRestrict_of_singleton hf hg] | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.CStarAlgebra.ApproximateUnit | {
"line": 57,
"column": 2
} | {
"line": 57,
"column": 78
} | [
{
"pp": "A : Type u_1\ninst✝² : NonUnitalCStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na b : A\nhab : a ≤ b\nha : 0 ≤ a\nhb : 0 ≤ b\n⊢ cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) a ≤ cfcₙ (fun x ↦ 1 - (1 + x)⁻¹) b",
"usedConstants": [
"cfcₙ",
"Eq.mpr",
"NonUnitalCStarAlgebra.toStarM... | rw [← inr_le_iff .., nnreal_cfcₙ_eq_cfc_inr a _, nnreal_cfcₙ_eq_cfc_inr b _] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Order | {
"line": 400,
"column": 2
} | {
"line": 400,
"column": 21
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : Aˣ\nha : 1 ≤ ↑a\n⊢ ↑a⁻¹ ≤ 1",
"usedConstants": [
"CStarAlgebra.inv_le_one"
]
}
] | exact inv_le_one ha | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.CStarAlgebra.Module.Constructions | {
"line": 352,
"column": 4
} | {
"line": 352,
"column": 53
} | [
{
"pp": "A : Type u_1\ninst✝³ : NonUnitalCStarAlgebra A\ninst✝² : PartialOrder A\nE : Type u_2\ninst✝¹ : NormedAddCommGroup E\ninst✝ : InnerProductSpace ℂ E\nx : E\n⊢ 0 ≤ ⟪x, x⟫_ℂ",
"usedConstants": [
"Eq.mpr",
"Real.instLE",
"Real",
"AddMonoid.toAddSemigroup",
"Inner.inner",
... | rw [← inner_self_ofReal_re, RCLike.ofReal_nonneg] | Lean.Parser.Tactic._aux_Init_Tactics___macroRules_Lean_Parser_Tactic_rwSeq_1 | Lean.Parser.Tactic.rwSeq |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 259,
"column": 2
} | {
"line": 263,
"column": 86
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝¹¹ : RCLike 𝕜\ninst✝¹⁰ : MeasurableSpace X\nμ : Measure X\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : StarRing A\ninst✝⁷ : NormedSpace 𝕜 A\ninst✝⁶ : IsScalarTower 𝕜 A A\ninst✝⁵ : SMulCommClass 𝕜 A A\ninst✝⁴ : NonUnitalContinuousFunctionalC... | refine integrable_cfcₙ' _ _ ⟨?_, ?_⟩ ha
· exact aeStronglyMeasurable_mkD_restrict_of_uncurry _ _ hf f_zero
· refine hasFiniteIntegral_mkD_restrict_of_bound f _ ?_ f_zero bound bound_int bound_ge
exact .of_forall fun x ↦
hf.comp (Continuous.prodMk_right x).continuousOn fun _ hz ↦ ⟨Set.mem_univ _, hz⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.ContinuousFunctionalCalculus.Integral | {
"line": 259,
"column": 2
} | {
"line": 263,
"column": 86
} | [
{
"pp": "X : Type u_1\n𝕜 : Type u_2\nA : Type u_3\np : A → Prop\ninst✝¹¹ : RCLike 𝕜\ninst✝¹⁰ : MeasurableSpace X\nμ : Measure X\ninst✝⁹ : NonUnitalNormedRing A\ninst✝⁸ : StarRing A\ninst✝⁷ : NormedSpace 𝕜 A\ninst✝⁶ : IsScalarTower 𝕜 A A\ninst✝⁵ : SMulCommClass 𝕜 A A\ninst✝⁴ : NonUnitalContinuousFunctionalC... | refine integrable_cfcₙ' _ _ ⟨?_, ?_⟩ ha
· exact aeStronglyMeasurable_mkD_restrict_of_uncurry _ _ hf f_zero
· refine hasFiniteIntegral_mkD_restrict_of_bound f _ ?_ f_zero bound bound_int bound_ge
exact .of_forall fun x ↦
hf.comp (Continuous.prodMk_right x).continuousOn fun _ hz ↦ ⟨Set.mem_univ _, hz⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.CStarMatrix | {
"line": 582,
"column": 25
} | {
"line": 582,
"column": 85
} | [
{
"pp": "m : Type u_1\nn : Type u_2\nA : Type u_5\ninst✝⁴ : Fintype m\ninst✝³ : NonUnitalCStarAlgebra A\ninst✝² : PartialOrder A\ninst✝¹ : StarOrderedRing A\ninst✝ : Fintype n\nM₁ M₂ : CStarMatrix m n A\n⊢ ‖M₁ + M₂‖ ≤ ‖M₁‖ + ‖M₂‖",
"usedConstants": [
"Pi.uniformSpace",
"Norm.norm",
"Semino... | by simpa [← map_add] using norm_add_le (toCLM M₁) (toCLM M₂) | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.InnerProductSpace.Dual | {
"line": 179,
"column": 91
} | {
"line": 181,
"column": 50
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : RCLike 𝕜\ninst✝² : NormedAddCommGroup E\ninst✝¹ : InnerProductSpace 𝕜 E\ninst✝ : CompleteSpace E\nx : E\ny : StrongDual 𝕜 E\n⊢ ⟪(toDual 𝕜 E).symm y, x⟫ = y x",
"usedConstants": [
"LinearIsometryEquiv.instEquivLike",
"Eq.mpr",
"InnerProduct... | by
rw [← toDual_apply_apply]
simp only [LinearIsometryEquiv.apply_symm_apply] | [anonymous] | Lean.Parser.Term.byTactic |
Mathlib.Analysis.Convex.Extreme | {
"line": 82,
"column": 52
} | {
"line": 82,
"column": 68
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA B : Set E\nx : E\nh : IsExtreme 𝕜 A B\ny z : E\nhx : x ∈ A\nhy : y ∈ A\nhz : z ∈ B\nhzxy : z ∈ openSegment 𝕜 x y\n⊢ z ∈ openSegment 𝕜 y x",
"usedConstants": [
"Eq.mpr... | openSegment_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.InnerProductSpace.Dual | {
"line": 234,
"column": 2
} | {
"line": 234,
"column": 67
} | [
{
"pp": "case hf\n𝕜 : Type u_3\nE : Type u_4\nF : Type u_5\ninst✝⁴ : RCLike 𝕜\ninst✝³ : SeminormedAddCommGroup E\ninst✝² : NormedSpace 𝕜 E\ninst✝¹ : NormedAddCommGroup F\ninst✝ : InnerProductSpace 𝕜 F\nx : E\ny : F\nhx : x ≠ 0\nhy : y ≠ 0\n⊢ (innerSL 𝕜) y ≠ 0",
"usedConstants": [
"LinearIsometry"... | · exact map_eq_zero_iff _ (toDualMap 𝕜 F).injective |>.not.mpr hy | Lean.Elab.Tactic.evalTacticCDot | Lean.cdot |
Mathlib.Analysis.Convex.Extreme | {
"line": 138,
"column": 7
} | {
"line": 138,
"column": 23
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : AddCommMonoid E\ninst✝ : SMul 𝕜 E\nA : Set E\nx : E\nh : x ∈ extremePoints 𝕜 A\nx₁ : E\nhx₁ : x₁ ∈ A\nx₂ : E\nhx₂ : x₂ ∈ A\nhx : x ∈ openSegment 𝕜 x₁ x₂\n⊢ x ∈ openSegment 𝕜 x₂ x₁",
"usedConstants": [
"E... | openSegment_symm | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Analysis.Convex.Extreme | {
"line": 215,
"column": 2
} | {
"line": 228,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_4\nM : ι → Type u_5\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (M i)\ninst✝ : (i : ι) → Module 𝕜 (M i)\ns : (i : ι) → Set (M i)\n⊢ extremePoints 𝕜 (univ.pi s) = univ.pi fun i ↦ extremePoints 𝕜 (s i)",
"usedConstants": [
"Set.e... | ext x
simp only [mem_extremePoints_iff_left, mem_univ_pi, @forall_and ι]
refine and_congr_right fun hx ↦ ⟨fun h i ↦ ?_, fun h ↦ ?_⟩
· rintro x₁ hx₁ x₂ hx₂ hi
rw [← update_self i x₁ x, h (update x i x₁) _ (update x i x₂)]
· rintro j
obtain rfl | hji := eq_or_ne j i <;> simp [*]
· rw [← Pi.image_u... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.Convex.Extreme | {
"line": 215,
"column": 2
} | {
"line": 228,
"column": 58
} | [
{
"pp": "𝕜 : Type u_1\nι : Type u_4\nM : ι → Type u_5\ninst✝³ : Semiring 𝕜\ninst✝² : PartialOrder 𝕜\ninst✝¹ : (i : ι) → AddCommGroup (M i)\ninst✝ : (i : ι) → Module 𝕜 (M i)\ns : (i : ι) → Set (M i)\n⊢ extremePoints 𝕜 (univ.pi s) = univ.pi fun i ↦ extremePoints 𝕜 (s i)",
"usedConstants": [
"Set.e... | ext x
simp only [mem_extremePoints_iff_left, mem_univ_pi, @forall_and ι]
refine and_congr_right fun hx ↦ ⟨fun h i ↦ ?_, fun h ↦ ?_⟩
· rintro x₁ hx₁ x₂ hx₂ hi
rw [← update_self i x₁ x, h (update x i x₁) _ (update x i x₂)]
· rintro j
obtain rfl | hji := eq_or_ne j i <;> simp [*]
· rw [← Pi.image_u... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 203,
"column": 14
} | {
"line": 203,
"column": 45
} | [
{
"pp": "X : Type u_1\nX' : Type u_2\nY : Type u_3\nY' : Type u_4\nZ : Type u_5\nZ' : Type u_6\ninst✝⁷ : TopologicalSpace X\ninst✝⁶ : TopologicalSpace X'\ninst✝⁵ : TopologicalSpace Y\ninst✝⁴ : TopologicalSpace Y'\ninst✝³ : TopologicalSpace Z\ninst✝² : TopologicalSpace Z'\ne✝ e e' : OpenPartialHomeomorph X Y\nin... | e.open_source.inter_frontier_eq | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.OpenPartialHomeomorph.Constructions | {
"line": 376,
"column": 17
} | {
"line": 376,
"column": 30
} | [
{
"pp": "case h\nX✝ : Type u_1\nX'✝ : Type u_2\nY : Type u_3\nY' : Type u_4\nZ✝ : Type u_5\nZ' : Type u_6\ninst✝⁹ : TopologicalSpace X✝\ninst✝⁸ : TopologicalSpace X'✝\ninst✝⁷ : TopologicalSpace Y\ninst✝⁶ : TopologicalSpace Y'\ninst✝⁵ : TopologicalSpace Z✝\ninst✝⁴ : TopologicalSpace Z'\ne✝ : OpenPartialHomeomorp... | ⟨x, hx, hx'x⟩ | Lean.Elab.Tactic.evalIntro | Lean.Parser.Term.anonymousCtor |
Mathlib.Analysis.CStarAlgebra.Matrix | {
"line": 202,
"column": 68
} | {
"line": 202,
"column": 87
} | [
{
"pp": "𝕜 : Type u_1\nm : Type u_2\nn : Type u_3\ninst✝³ : RCLike 𝕜\ninst✝² : Fintype m\ninst✝¹ : Fintype n\ninst✝ : DecidableEq n\nA : Matrix m n 𝕜\n⊢ ‖toContinuousLinearMap\n ((toLin (EuclideanSpace.basisFun m 𝕜).toBasis (EuclideanSpace.basisFun n 𝕜).toBasis) Aᴴ ∘ₗ\n (toLin (EuclideanSpa... | toLin_conjTranspose | Lean.Elab.Tactic.evalRewriteSeq | null |
Mathlib.Topology.FiberBundle.Basic | {
"line": 353,
"column": 2
} | {
"line": 353,
"column": 88
} | [
{
"pp": "case inr\nB : Type u_2\nF : Type u_3\ninst✝⁶ : TopologicalSpace B\ninst✝⁵ : TopologicalSpace F\nE : B → Type u_5\ninst✝⁴ : TopologicalSpace (TotalSpace F E)\ninst✝³ : (b : B) → TopologicalSpace (E b)\ninst✝² : ConditionallyCompleteLinearOrder B\ninst✝¹ : OrderTopology B\ninst✝ : FiberBundle F E\na b : ... | set s : Set B := { x ∈ Icc a b | ∃ e : Trivialization F (π F E), Icc a x ⊆ e.baseSet } | Mathlib.Tactic._aux_Mathlib_Tactic_Set___elabRules_Mathlib_Tactic_setTactic_1 | Mathlib.Tactic.setTactic |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 623,
"column": 18
} | {
"line": 623,
"column": 52
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nB' : Type u_5\ninst✝ : TopologicalSpace B'\nh : B ≃ₜ B'\n⊢ (e.tr... | ext; simp [Prod.map, e.mem_target] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 623,
"column": 18
} | {
"line": 623,
"column": 52
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace F E)\ne : Trivialization F proj\nx : Z\nB' : Type u_5\ninst✝ : TopologicalSpace B'\nh : B ≃ₜ B'\n⊢ (e.tr... | ext; simp [Prod.map, e.mem_target] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 730,
"column": 18
} | {
"line": 730,
"column": 64
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace F E)\ne✝ : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nb : B\ny : E b\nF' : Typ... | simp [target_eq, prod_univ, preimage_preimage] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 730,
"column": 18
} | {
"line": 730,
"column": 64
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace F E)\ne✝ : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nb : B\ny : E b\nF' : Typ... | simp [target_eq, prod_univ, preimage_preimage] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Topology.FiberBundle.Trivialization | {
"line": 730,
"column": 18
} | {
"line": 730,
"column": 64
} | [
{
"pp": "B : Type u_1\nF : Type u_2\nE : B → Type u_3\nZ : Type u_4\ninst✝⁴ : TopologicalSpace B\ninst✝³ : TopologicalSpace F\nproj : Z → B\ninst✝² : TopologicalSpace Z\ninst✝¹ : TopologicalSpace (TotalSpace F E)\ne✝ : Trivialization F proj\nx : Z\ne' : Trivialization F TotalSpace.proj\nb : B\ny : E b\nF' : Typ... | simp [target_eq, prod_univ, preimage_preimage] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Multiplier | {
"line": 614,
"column": 6
} | {
"line": 614,
"column": 45
} | [
{
"pp": "𝕜 : Type u_1\nA : Type u_2\ninst✝⁸ : DenselyNormedField 𝕜\ninst✝⁷ : StarRing 𝕜\ninst✝⁶ : NonUnitalNormedRing A\ninst✝⁵ : StarRing A\ninst✝⁴ : CStarRing A\ninst✝³ : NormedSpace 𝕜 A\ninst✝² : SMulCommClass 𝕜 A A\ninst✝¹ : IsScalarTower 𝕜 A A\ninst✝ : StarModule 𝕜 A\na : 𝓜(𝕜, A)\nhball : (Metric.... | simp only [← @opNNNorm_mul_apply 𝕜 _ A] | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 71,
"column": 2
} | {
"line": 74,
"column": 11
} | [
{
"pp": "x y : ℝ\n⊢ exp x = exp y ↔ ∃ m, x = y + ↑m * (2 * π)",
"usedConstants": [
"add_mul",
"Int.cast",
"Eq.mpr",
"Complex.exp_eq_exp_iff_exists_int",
"NormedCommRing.toSeminormedCommRing",
"Semigroup.toMul",
"Real",
"Real.pi",
"HMul.hMul",
"cong... | rw [Subtype.ext_iff, coe_exp, coe_exp, exp_eq_exp_iff_exists_int]
refine exists_congr fun n => ?_
rw [← mul_assoc, ← add_mul, mul_left_inj' I_ne_zero]
norm_cast | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.SpecialFunctions.Complex.Circle | {
"line": 71,
"column": 2
} | {
"line": 74,
"column": 11
} | [
{
"pp": "x y : ℝ\n⊢ exp x = exp y ↔ ∃ m, x = y + ↑m * (2 * π)",
"usedConstants": [
"add_mul",
"Int.cast",
"Eq.mpr",
"Complex.exp_eq_exp_iff_exists_int",
"NormedCommRing.toSeminormedCommRing",
"Semigroup.toMul",
"Real",
"Real.pi",
"HMul.hMul",
"cong... | rw [Subtype.ext_iff, coe_exp, coe_exp, exp_eq_exp_iff_exists_int]
refine exists_congr fun n => ?_
rw [← mul_assoc, ← add_mul, mul_left_inj' I_ne_zero]
norm_cast | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.CStarAlgebra.Unitary.Connected | {
"line": 166,
"column": 2
} | {
"line": 166,
"column": 44
} | [
{
"pp": "case a\nA : Type u_1\ninst✝ : CStarAlgebra A\nx : ↥(selfAdjoint A)\nhx : ‖x‖ < π\na✝ : Nontrivial A\nthis : spectrum ℂ ↑(expUnitary x) ⊆ slitPlane\ny : ℝ\nhy : ‖y‖ < π\n⊢ (cexp (↑y * I)).arg = y",
"usedConstants": [
"AddGroup.toSubtractionMonoid",
"Norm.norm",
"NegZeroClass.toNeg"... | simp only [Real.norm_eq_abs, abs_lt] at hy | Lean.Elab.Tactic.evalSimp | Lean.Parser.Tactic.simp |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 38,
"column": 2
} | {
"line": 52,
"column": 89
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : IsSelfAdjoint a\nha_norm : ‖a‖ ≤ 1\n⊢ a + I • CFC.sqrt (1 - a ^ 2) ∈ unitary A",
"usedConstants": [
"IsSelfAdjoint.sq_nonneg",
"Nontrivial",
"cfc_pow",
"AddGroup.toSubtract... | obtain (_ | _) := subsingleton_or_nontrivial A
· simp [Subsingleton.elim (a + I • CFC.sqrt (1 - a ^ 2)) 1, one_mem (unitary A)]
have key : a + I • CFC.sqrt (1 - a ^ 2) = cfc (fun x : ℂ ↦ x.re + I * √(1 - x.re ^ 2)) a := by
rw [CFC.sqrt_eq_real_sqrt (1 - a ^ 2) ?nonneg]
case nonneg =>
rwa [sub_nonneg, ... | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.CStarAlgebra.Unitary.Span | {
"line": 38,
"column": 2
} | {
"line": 52,
"column": 89
} | [
{
"pp": "A : Type u_1\ninst✝² : CStarAlgebra A\ninst✝¹ : PartialOrder A\ninst✝ : StarOrderedRing A\na : A\nha : IsSelfAdjoint a\nha_norm : ‖a‖ ≤ 1\n⊢ a + I • CFC.sqrt (1 - a ^ 2) ∈ unitary A",
"usedConstants": [
"IsSelfAdjoint.sq_nonneg",
"Nontrivial",
"cfc_pow",
"AddGroup.toSubtract... | obtain (_ | _) := subsingleton_or_nontrivial A
· simp [Subsingleton.elim (a + I • CFC.sqrt (1 - a ^ 2)) 1, one_mem (unitary A)]
have key : a + I • CFC.sqrt (1 - a ^ 2) = cfc (fun x : ℂ ↦ x.re + I * √(1 - x.re ^ 2)) a := by
rw [CFC.sqrt_eq_real_sqrt (1 - a ^ 2) ?nonneg]
case nonneg =>
rwa [sub_nonneg, ... | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 214,
"column": 26
} | {
"line": 214,
"column": 59
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝³ : DivisionRing k\ninst✝² : AddCommGroup V\ninst✝¹ : Module k V\ninst✝ : AffineSpace V P\nm n : ℕ\ns : Simplex k P m\ne : Fin (m + 1) ≃ Fin (n + 1)\n⊢ m = n",
"usedConstants": [
"Iff.mpr",
"Fintype.card_fin",
"congrArg",
"Eq.mp... | simpa using Fintype.card_eq.2 ⟨e⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.AffineSpace.FiniteDimensional | {
"line": 100,
"column": 2
} | {
"line": 101,
"column": 99
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\nι : Type u_4\ninst✝⁴ : DivisionRing k\ninst✝³ : AddCommGroup V\ninst✝² : Module k V\ninst✝¹ : AffineSpace V P\ninst✝ : FiniteDimensional k V\np : ι → P\na✝ : Nontrivial ι\ninhabited_h : Inhabited ι\nhi : LinearIndependent k fun i ↦ p ↑i -ᵥ p default\nthis : IsN... | exact
(Set.finite_singleton default).finite_of_compl (Set.finite_coe_iff.1 hi.finite_of_isNoetherian) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 165,
"column": 2
} | {
"line": 167,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nB : F →L[𝕜] E\n⊢ A = adjoint B ↔ ∀ (x : E) (... | refine ⟨fun h x y => by rw [h, adjoint_inner_left], fun h => ?_⟩
ext x
exact ext_inner_right 𝕜 fun y => by simp only [adjoint_inner_left, h x y] | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 416,
"column": 27
} | {
"line": 416,
"column": 60
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nm n : ℕ\ninst✝¹ : NeZero m\ninst✝ : NeZero n\ns : Simplex k P m\ne : Fin (m + 1) ≃ Fin (n + 1)\ni : Fin (n + 1)\n⊢ m = n",
"usedConstants": [
"Iff.mpr",
... | simpa using Fintype.card_eq.2 ⟨e⟩ | Lean.Elab.Tactic.Simpa.evalSimpa | Lean.Parser.Tactic.simpa |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 416,
"column": 27
} | {
"line": 416,
"column": 60
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nm n : ℕ\ninst✝¹ : NeZero m\ninst✝ : NeZero n\ns : Simplex k P m\ne : Fin (m + 1) ≃ Fin (n + 1)\ni : Fin (n + 1)\n⊢ m = n",
"usedConstants": [
"Iff.mpr",
... | simpa using Fintype.card_eq.2 ⟨e⟩ | Lean.Elab.Tactic.evalTacticSeq1Indented | Lean.Parser.Tactic.tacticSeq1Indented |
Mathlib.Analysis.InnerProductSpace.Adjoint | {
"line": 165,
"column": 2
} | {
"line": 167,
"column": 75
} | [
{
"pp": "𝕜 : Type u_1\nE : Type u_2\nF : Type u_3\ninst✝⁶ : RCLike 𝕜\ninst✝⁵ : NormedAddCommGroup E\ninst✝⁴ : NormedAddCommGroup F\ninst✝³ : InnerProductSpace 𝕜 E\ninst✝² : InnerProductSpace 𝕜 F\ninst✝¹ : CompleteSpace E\ninst✝ : CompleteSpace F\nA : E →L[𝕜] F\nB : F →L[𝕜] E\n⊢ A = adjoint B ↔ ∀ (x : E) (... | refine ⟨fun h x y => by rw [h, adjoint_inner_left], fun h => ?_⟩
ext x
exact ext_inner_right 𝕜 fun y => by simp only [adjoint_inner_left, h x y] | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.LinearAlgebra.AffineSpace.Simplex.Centroid | {
"line": 416,
"column": 27
} | {
"line": 416,
"column": 60
} | [
{
"pp": "k : Type u_1\nV : Type u_2\nP : Type u_3\ninst✝⁵ : DivisionRing k\ninst✝⁴ : AddCommGroup V\ninst✝³ : Module k V\ninst✝² : AffineSpace V P\nm n : ℕ\ninst✝¹ : NeZero m\ninst✝ : NeZero n\ns : Simplex k P m\ne : Fin (m + 1) ≃ Fin (n + 1)\ni : Fin (n + 1)\n⊢ m = n",
"usedConstants": [
"Iff.mpr",
... | simpa using Fintype.card_eq.2 ⟨e⟩ | Lean.Elab.Tactic.evalTacticSeq | Lean.Parser.Tactic.tacticSeq |
Mathlib.Analysis.Normed.Affine.AddTorsorBases | {
"line": 94,
"column": 4
} | {
"line": 94,
"column": 57
} | [
{
"pp": "V : Type u_1\nP : Type u_2\ninst✝³ : NormedAddCommGroup V\ninst✝² : NormedSpace ℝ V\ninst✝¹ : MetricSpace P\ninst✝ : NormedAddTorsor V P\ns u : Set P\nhu : IsOpen u\nhsu : s ⊆ u\nh : AffineIndependent ℝ Subtype.val\nq : P\nhq : q ∈ s\nε : ℝ\nε0 : 0 < ε\nhεu : Metric.closedBall q ε ⊆ u\nt : Set P\nht₁ :... | exact mul_le_of_le_one_left ε0.le (div_self_le_one _) | Lean.Elab.Tactic.evalExact | Lean.Parser.Tactic.exact |
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